Summary of Quantum and Statistical Mechanics

John von Neumann

This introductory chapter provides a brief summary of the discussions in the succeeding chapters. Here, the new quantum mechanics has in recent years achieved in its essential parts what is presumably a definitive form: the so-called “transformation theory.” Therefore the principal emphasis shall be placed on the general and fundamental questions which have arisen in connection with this theory. In particular, the difficult problems of interpretation, many of which are even now not fully resolved, will be investigated in detail. In this connection, the relation of quantum mechanics to statistics and to the classical statistical mechanics is of special importance.

G. E. Uhlenbeck

Elias P. Gyftopoulos

The prevailing beliefs in the scientific and engineering literature are that: (i) thermodynamics is explained and justified by statistical mechanics; (ii) entropy is a statistical measure of disorder; and (iii) for given values of energy, volume, and amounts of constituents, the largest value of entropy corresponds to both a thermodynamic equilibrium state and the ultimate disorder. In this paper, we provide: (i) a summary of the beliefs as stated by some eminent scientists; (ii) experimental evidence that casts serious doubt about the validity of the beliefs; (iii) an outline of a nonstatistical unified quantum theory of mechanics and thermodynamics; (iv) an outline of a nonquantal, nonstatistical exposition of thermodynamics, valid for all systems (both macroscopic and microscopic), and for all states (both thermodynamic equilibrium and not thermodynamic equilibrium); (v) the definition and analytical expression of the entropy of thermodynamics; (vi) the interpretation of entropy as both a measure of the quantum-theoretic spatial shape of a molecule, and an indicator of order; and (vii) nonstatistical answers to the questions that motivated the introduction of statistical mechanics.

Peter L. Hagelstein, Stephen D. Senturia, Terry P. Orlando

D. ter Haar

Summary A brief account is given of the reasons why the many-body problem has recently become the subject of a rapidly expanding literature. After a survey of the main ideas of recent theories, it is shown how these ideas have played a rôle in giving us a better understanding of such topics as superconductivity, plasmas, and liquid helium. In the theory of the many-body problem the main branches of theoretical physics applied are Hamiltonian mechanics or quantum mechanics (to describe the motion of the constituent particles of the many-body system under consideration) and statistical mechanics (to describe the collective action of the large number of particles in the system). As these subjects are relatively advanced we have discussed in an appendix the main concepts of Hamiltonian mechanics and some of those of quantum statistics, including the definition and some of the properties of fermions and bosons.